This is one of the trickier word differences that can end up tripping people up a bit in their business writing. That can be frustrating as they have very different meanings and can result in you saying something very different from what you mean.
To compliment someone, with an "i", is the most common usage and means to give praise. "When I said that to you I meant it as a compliment", or "my compliments to the chef," are two ways you would use this.

To complement refers to supporting or enhancing something and is the usage you would probably see more in a business context. "Susie has done a good job of complementing the HR Department," or "acquiring that company would be a great complement to our existing capabilities". If you used compliment with an "i" in Susie's context think about what you would be saying here. That Suzy has done a good job of giving praise to the HR group would somewhat defeat the purpose when you're trying to comment on how she's a great addition to the HR Department.

Keeping the meanings of these two straight is important, as this can completely change the message that you are trying to convey.